Modeling and Valuation of Energy Structures Applied Quantitative Finance series Applied Quantitative Finance is a new series developed to bring readers the very latest market tested tools, techniques and developments in quantitative finance Written for practitioners who need to understand how things work “on the floor”, the series will deliver the most cutting-edge applications in areas such as asset pricing, risk management and financial derivatives Although written with practitioners in mind, this series will also appeal to researchers and students who want to see how quantitative finance is applied in practice Also available Oliver Brockhaus EQUITY DERIVATIVES AND HYBRIDS Markets, Models and Methods Enrico Edoli, Stefano Fiorenzani and Tiziano Vargiolu OPTIMIZATION METHODS FOR GAS AND POWER MARKETS Theory and Cases Roland Lichters, Roland Stamm and Donal Gallagher MODERN DERIVATIVES PRICING AND CREDIT EXPOSURE ANALYSIS Theory and Practice of CSA and XVA Pricing, Exposure Simulation and Backtesting Zareer Dadachanji FX BARRIER OPTIONS A Comprehensive Guide for Industry Quants Ignacio Ruiz XVA DESKS: A NEW ERA FOR RISK MANAGEMENT Understanding, Building and Managing Counterparty and Funding Risk Christian Crispoldi, Peter Larkin and Gérald Wigger SABR AND SABR LIBOR MARKET MODEL IN PRACTICE With Examples Implemented in Python Adil Reghai QUANTITATIVE FINANCE Back to Basic Principles Chris Kenyon and Roland Stamm DISCOUNTING, LIBOR, CVA AND FUNDING Interest Rate and Credit Pricing Marc Henrard INTEREST RATE MODELLING IN THE MULTI-CURVE FRAMEWORK Foundations, Evolution and Implementation Modeling and Valuation of Energy Structures Analytics, Econometrics, and Numerics Daniel Mahoney Director of Quantitative Analysis, Citigroup, USA © Daniel Mahoney 2016 All rights reserved No reproduction, copy or transmission of this publication may be made without written permission No portion of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, Saffron House, 6–10 Kirby Street, London EC1N 8TS Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages The author has asserted his right to be identified as the author of this work in accordance with the Copyright, Designs and Patents Act 1988 First published 2016 by PALGRAVE MACMILLAN Palgrave Macmillan in the UK is an imprint of Macmillan Publishers Limited, registered in England, company number 785998, of Houndmills, Basingstoke, Hampshire RG21 6XS Palgrave Macmillan in the US is a division of St Martin’s Press LLC, 175 Fifth Avenue, New York, NY 10010 Palgrave Macmillan is the global academic imprint of the above companies and has companies and representatives throughout the world Palgrave® and Macmillan® are registered trademarks in the United States, the United Kingdom, Europe and other countries ISBN: 978–1–137–56014–8 This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources Logging, pulping and manufacturing processes are expected to conform to the environmental regulations of the country of origin A catalogue record for this book is available from the British Library A catalog record for this book is available from the Library of Congress To Cathy, Maddie, and Jack Contents List of Figures List of Tables Preface Acknowledgments Synopsis of Selected Energy Markets and Structures 1.1 Challenges of modeling in energy markets 1.1.1 High volatilities/jumps 1.1.2 Small samples 1.1.3 Structural change 1.1.4 Physical/operational constraints 1.2 Characteristic structured products 1.2.1 Tolling arrangements 1.2.2 Gas transport 1.2.3 Gas storage 1.2.4 Load serving 1.3 Prelude to robust valuation Data Analysis and Statistical Issues 2.1 Stationary vs non-stationary processes 2.1.1 Concepts 2.1.2 Basic discrete time models: AR and VAR 2.2 Variance scaling laws and volatility accumulation 2.2.1 The role of fundamentals and exogenous drivers 2.2.2 Time scales and robust estimation 2.2.3 Jumps and estimation issues 2.2.4 Spot prices 2.2.5 Forward prices 2.2.6 Demand side: temperature 2.2.7 Supply side: heat rates, spreads, and production structure 2.3 A recap Valuation, Portfolios, and Optimization 3.1 Optionality, hedging, and valuation 3.1.1 Valuation as a portfolio construction problem 3.1.2 Black Scholes as a paradigm 3.1.3 Static vs dynamic strategies 3.1.4 More on dynamic hedging: rolling intrinsic 3.1.5 Market resolution and liquidity 3.1.6 Hedging miscellany: greeks, hedge costs, and discounting 3.2 Incomplete markets and the minimal martingale measure 3.2.1 Valuation and dynamic strategies 3.2.2 Residual risk and portfolio analysis 3.3 Stochastic optimization 3.3.1 Stochastic dynamic programming and HJB 3.3.2 Martingale duality 3.4 Appendix 3.4.1 Vega hedging and value drivers 3.4.2 Value drivers and information conditioning Selected Case Studies 4.1 Storage 4.2 Tolling 4.3 Tolling 4.3.1 (Monthly) Spread option representation of storage 4.3.2 Lower-bound tolling payoffs Analytical Techniques 5.1 Change of measure techniques 5.1.1 Review/main ideas 5.1.2 Dimension reduction/computation facilitation/estimation robustness 5.1.3 Max/min options 5.1.4 Quintessential option pricing formula 5.1.5 Symmetry results: Asian options 5.2 Affine jump diffusions/characteristic function methods 5.2.1 Lévy processes 5.2.2 Stochastic volatility 5.2.3 Pseudo-unification: affine jump diffusions 5.2.4 General results/contour integration 5.2.5 Specific examples 5.2.6 Application to change of measure 5.2.7 Spot and implied forward models 5.2.8 Fundamental drivers and exogeneity 5.2.9 Minimal martingale applications 5.3 Appendix 5.3.1 More Asian option results 5.3.2 Further change-of-measure applications Econometric Concepts 6.1 Cointegration and mean reversion 6.1.1 Basic ideas 6.1.2 Granger causality 6.1.3 Vector Error Correction Model (VECM) 6.1.4 Connection to scaling laws 6.2 Stochastic filtering 6.2.1 Basic concepts 6.2.2 The Kalman filter and its extensions 6.2.3 Heston vs generalized autoregressive conditional heteroskedasticity (GARCH) 6.3 Sampling distributions 6.3.1 The reality of small samples 6.3.2 Wishart distribution and more general sampling distributions 6.4 Resampling and robustness 6.4.1 Basic concepts 6.4.2 Information conditioning 6.4.3 Bootstrapping 6.5 Estimation in finite samples 6.5.1 Basic concepts 6.5.2 MLE and QMLE 6.5.3 GMM, EMM, and their offshoots 6.5.4 A study of estimators in small samples 6.5.5 Spectral methods 6.6 Appendix 6.6.1 Continuous vs discrete time 6.6.2 Estimation issues for variance scaling laws 6.6.3 High-frequency scaling Numerical Methods 7.1 Basics of spread option pricing 7.1.1 Measure changes 7.1.2 Approximations 7.2 Conditional expectation as a representation of value 7.3 Interpolation and basis function expansions 7.3.1 Pearson and related approaches 7.3.2 The grid model 7.3.3 Further applications of characteristic functions 7.4 Quadrature 7.4.1 Gaussian 7.4.2 High dimensions 7.5 Simulation 7.5.1 Monte Carlo 7.5.2 Variance reduction 7.5.3 Quasi-Monte Carlo 7.6 Stochastic control and dynamic programming 7.6.1 Hamilton-Jacobi-Bellman equation 7.6.2 Dual approaches 7.6.3 LSQ 7.6.4 Duality (again) 7.7 Complex variable techniques for characteristic function applications 7.7.1 Change of contour/change of measure 7.7.2 FFT and other transform methods Dependency Modeling 8.1 Dependence and copulas 8.1.1 Concepts of dependence 8.1.2 Classification 8.1.3 Dependency: continuous vs discontinuous processes 8.1.4 Consistency: static vs dynamic 8.1.5 Wishart processes 8.2 Signal and noise in portfolio construction 8.2.1 Random matrices 8.2.2 Principal components and related concepts Notes Bibliography Index Journal of 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processes, 22–9 estimation of, 22–3 scalar, 22–6 vector, 26–9 Bachelier model, see Gaussian options basis expansions grid-based quadrature, 279–304 in simulation, 339–44 behavior, 363–5 simulation, 363–4 Sklar’s theorem, 360, 379–80 tail dependence, 362–3 vine, 370–1 Black-Scholes model, 52–8 central assumptions of, 52–3 key insights of, 53–6 bootstrapping, 235–6 as a stability test, 236 cash volatility, 9–10, 125–8 dependence on market resolution, 65–8 central limit theorem (CLT), 237–8, 320, 333 change of measure, 131–45 computational benefits, 135–42, 346–53 econometric relevance, 81, 138 in option pricing, 57–8 in simulation, 324–5 see also minimal martingale measure characteristic functions, 145–57, 353–8 Cholesky decomposition in quadrature, 313–14 in simulation, 322–3 cointegration, 191–207 common stochastic drivers, 192–3 Granger causality, 197–9 Johansen’s eigenvalue test, 201–5 long-term correlation, 197 and OLS, 196 spurious regressions, 193–6 stochastic trends, 198–9 and variance scaling laws, 205–7 vector error correction model, 199–205 continuous time, 258–60 contour integration, 157–66 relation to change of measure, 346–53 convergence of estimators, 23–6, 237–54 in simulation, 319–20, 328–37 copulas, 359–81 Archimedean, 365 cross-commodity/cross-asset, 364–5 dynamics, 374–81 elliptical, 366–8 empirical, 369 generalized elliptical, 368–9 Lévy, 372–4, 376–81 measures of dependency, 359–63 product, 369–70 separation of joint and marginal correlation, appropriateness of as dependence measure, 360–1 as valuation parameter, 78–9, 80–1, 126 Cox-Ingersoll-Ross (CIR), 237, 352 see also Heston model crude oil, 39–43 financialization of, 41–2 diagnostics, see econometrics diffusion process, 145–6, 374–5 discounting, 84–5 discrete time, 258–60 drift, process in commodity markets, 58–64 see also mean reversion duality in optimization, 106–7 in simulation, 107–11, 337–46 dynamic programming and early exercise options, 293–300 and stochastic control, 101–6, 337–8 econometrics diagnosis in, 16–22 limitations of asymptotic results, 23–6, 237–54 objectives for hedging and valuation, 16 see also specific techniques efficient method of moments (EMM), 247 eigenvalues, 202–4, 313–15 role in variance scaling laws, 205–7 estimation objectives of, 12–19 see also econometrics estimators, 19–22, 231–5 expectations conditional vs unconditional, 14–17 and valuation, 279 fast Fourier transform (FFT), 287, 291, 353–8 computational benefits of, 288 fractional, 355–6 see also Fourier methods filtering, 207–20 challenges of, 208–9 Markov chain Monte Carlo, 209 Nonlinear, 214–16 see also Kalman filtering forward models, 51–2, 118–30 relation to spot models, 119–21, 169–74 forward prices, 3, 40–3, 76 forward volatility, see monthly volatility Fourier methods in econometrics, 255–8 in option pricing, 157–60, 353–8 see also characteristic functions full requirements, 9–11 fundamental drivers, 31–6, 191 capital formation effects, 29, 31, 164 exogeneity, 174–8 role in price formation, 32–3 role in valuation, 61–7 supply and demand, 46–7 futures prices, see forward prices Gaussian options and characteristic function applications, 159–60, 161–2 relevance for natural gas transport, 6–7 generalized method of moments (GMM), 244–6 geometric brownian motion (GBM), 52–4, 59–60, 69, 161 Girsanov’s Theorem, 133–5, 325 greeks, 79–83, 117, 137–9, 288–90, 307 delta, 53–5, 62, 64, 73–4, 79, 93–6, 114–15, 181–2, 283 finite difference vs simulation, 328–33 gamma, 54, 80, 84–5, 92, 95, 283 and hedging, 48–50 vega, 10, 17, 79, 94–5, 111–13, 115, 126–7, 182, 276, 283 Hamilton-Jacobi-Bellman (HJB) equation, 104, 338 heat rates models, 31–6, 61–3 variance scaling law of, 46–7 hedging as counterpart of valuation, 48–58 dynamic vs static, 58–68 proxy, 12–14 as transformation of risk, 50 Heston model, 93–101, 113–17, 151–3, 163–4, 302–4, 306 generalized, 180–1, 188–9, 382 high-dimensional problems, 313–18 see also simulation hypothesis testing, 17–18, 23 lack of relevance, 17, 47 indirect inference, 246–7 information accumulation, 29–34 relation to variance scaling, 29–30 inverse leverage effect, 149 Itô’s lemma, 80, 91, 104, 137 jumps, 1–2, 145–56, 162–3, 190, 375 bipower variation, 270–1 estimation issues, 34–6 high frequency, 268–71 and measure change, 134–5 Kalman filtering, 209–20 continuous-time limit, 216–17 extended, 214 performance of, 218–20 unscented, 214–16 Karush-Kuhn-Tucker (KKT) condition, 106–7 and Lagrange multipliers, 106 see also duality Kirk’s approximation, see spread options least squares Monte Carlo (LSQ), 127, 339–44 Lévy processes, 145–8 Lévy-Khintchine representation, 146, 372 and stochastic volatility, 149–54 linear programming, 128–9 liquidity, 12–14, 51–2, 56, 75–9 load serving, see full requirements local time, 68–75 connection with rolling intrinsic strategy, 71–5 Tanaka-Meyer formula, 69–70 log-likelihood function, 21–2, 201–3, 208, 212, 238–40 Margrabe formula, 62, 72, 131, 138, 272–4 markets, conceptual complete vs incomplete, 85–101 efficient, 13, 33, 42 markets, energy geographical segmentation, jumps, 2–3 physical and operational constraints, structural change, volatility, 2–3 see also liquidity Markov property, 103, 114, 209, 338 martingales, 42, 49, 57–8, 85–93, 107–9, 113–17, 123–7, 131–7, 145, 147, 150, 178–81, 268–70, 279, 339, 344–6 maximum likelihood estimation (MLE), 21–2, 238–42 mean reversion impact on variance estimation, 36–9, 260–8 reflected in variance scaling laws, 29–30, 39–47, 59–60 see also time scales measure (probability) change of, 131–5, 143–5, 158–9, 272–5, 324–5, 346–53 equivalent, 88–9, 131–3 and martingales, 88–9 physical, 52, 54, 62, 88, 1701–1 pricing, 49–50, 57–8, 65, 86–7, 170–1 Merton jump diffusion, 162–3 minimal martingale measure, 85–99, 178–84 entropy minimization, 182–4 optimal hedging, 95–9 and variance minimization, 181–2 Monte Carlo, see simulation monthly volatility, 9, 64–5, 77, 126 natural gas, 3, 39–43 basis, 6–7 storage, 7–8, 71–5, 79, 102–3, 109–11, 118–21, 128–9, 139, 171, 296–300, 307 transportation, 6–7, 79 non-arbitrage, 53–4, 56, 68–71 limits of, 85–95 nonstationarity, 12–18, 23–4, 28–9, 37, 39–43, 164–6, 191 numeraire, 57, 65, 136–7, 139, 143–4, 279 operational constraints, see also storage; tolling optimization, 106–7 see also Hamilton-Jacobi-Bellman (HJB) equation option valuation, 48–58, 157–66 options, 4–8 American, 101–5, 107–8, 293–6, 302–4, 339 European, 53 max/min, 139–40 moneyness, 64, 65–7, 76 spread, 61–4, 71–2, 80–1, 111–13, 128–30, 135–9, 272–8, 279–84, 331–3, 356–8 with multiple exercise rights, 299–300, 344–6 ordinary differential equation (ODE) in option valuation, 155–6 ordinary least squares (OLS), 19–21, 22, 193–6, 245 outages, 5, 31–3, 35–6, 86, 122, 150, 190 pathwise relationships, 10, 16–17, 50, 58 payoff functions, 48–9 Pearson’s method, see spread options Poisson process, 134–5, 146–8, 155, 271, 372–3, 375, 378–9 population, 10, 18, 37, 225, 232 vs sample, 13, 16, 18–19, 20, 22, 33–4, 194, 237–9, 244, 254, 261 portfolios as essence of valuation, 48–85 and random matrices, 384–6 signal and noise issues, 386–9 and variance minimization, 85–101 principal components analysis (PCA), 327, 389–90 process modeling, 154, 374–5 pure jump process, 147–8, 376–7 quadratic variation, 41–2, 59–60, 64–8, 77, 268–71 quadrature, 279–318 comparison with binomial trees, 294–6 Gaussian, 305–13 simulation and, 319–20 sparse grid, 315–18 quasi-maximum likelihood estimation (QMLE), 242–4 Kullback-Leibler (KL) divergence, 243 quasi-Monte Carlo, 333–7 clustering issues, 335–6 low-discrepancy sequences, 333–4 Sobol’, 335 Radon-Nikokym derivative, 131–2, 178–80 random matrices, 384–8 recalls, see swing options regressions in estimation, 19–21, 22–3 in simulation, 340–3 see also cointegration residuals as sample entity, 13–14 see also risk risk adjustment, 13, 36, 71, 81, 84, 241–2 residual, 10–11, 14, 16–17, 50–1, 87, 88–93, 95–6, 115–17 risk neutrality irrelevance of, 94–5 meaning of, 57 robustness and econometric stability, 16, 25, 30, 33–4, 47, 138, 231–6 vs structure, 4–5, 11, 49, 74, 87, 92 rolling intrinsic, 68–75, 83 sample impact of finite size, 2–3, 16, 20, 22–4, 33–4, 37–8, 192, 193–4, 225, 232–3, 247–54 vs population, 13, 16, 18–19, 20, 22, 33–4, 194, 237–9, 244, 254, 261 sampling distribution, 33–4, 225–31 Samuelson effect, see volatility term structure Schwartz model, 164–6 seasonality, 7–8, 43–4 shadow price, 105, 110–11 simulation, 23–6, 73–5, 96–9, 122–7, 204, 209, 219, 247–54, 318–37 Brownian bridge, 325–7 generation of random deviates, 320–1 importance sampling, 324–5 and information processing, 318–19 likelihood ratio, 328–33 pseudo-random, 320 quasi-Monte Carlo, 333–7 variance reduction, 323–7 smile, volatility, 149 spark spread option, 5, 61–4, 79, 85 spectral methods in estimation, 255–8 see also characteristic functions spot models relation to forward models, 51–2 spot prices, 39–42 spot volatility, see cash volatility spread options, 272–85 Kirk’s approximation, 276–8 Pearson’s method, 280–5 stationarity, 12–18, 29–30, 43–7, 145–6, 164–6, 240–7 statistical significance, 13–14, 16–18, 21, 25, 196, 234–6 stochastic control, 8, 101–11, 118–19, 122–3, 218, 296–7, 337–46 stochastic dynamic programming, see stochastic control stochastic volatility, 148, 149 representations of, 149–54 see also Heston model storage, natural gas, 7–8, 71–6, 79, 102–3 lower bound valuation, 118–21, 128–9, 296–8 upper bound valuation, 109–11 swing option, 299–300 Tanaka-Meyer, 69–70 temperature as a fundamental driver, 43–4, 164, 175 non-stationary effects in, 34, 46 stationarity of, 14–15, 32, 44 variance scaling law of, 44–6 time changes, stochastic, 149–54, 372 Laplace transforms, 150 time scales, 1, 14–16, 18, 30, 31–2, 33–5, 37, 40, 45, 78, 166, 177–8, 197, 199, 222–3, 267–8 tolling, 4–5, 75, 77, 102–3, 121–8, 139 and load-serving, 10 lower bound valuation, 125–6, 128–9, 307–9 upper bound valuation, 123–5, 125–6 transaction costs, 64, 70, 78, 83–4 bid-ask spread, 74, 76, 78 transform methods, see characteristic functions transportation, natural gas, 6–7, 77, 79, 84, 139, 165, 274 valuation absolute vs relative pricing, 11, 57, 88–9, 94 arbitrage arguments, 52–4, 93–5 value drivers, 50–1, 54–6, 62, 70–1, 74, 78–9, 79–82, 90–1, 111–17, 225 variance estimation of, 24–6 realized, 93, 95, 55, 74, 115 scaling, 18, 29–47, 176–8 vs quadratic variation, 58–68, 172–4 vector autoregression (VAR), 26–9, 199–200 volatility implied, 54, 76, 112–13, 149 term structure, 15, 32, 41–2, 165–6, 173–4, 177, 189 weather, see temperature Wishart distribution, 226–31, 381–3 extension to non-Gaussian case, 230–1 and sampling distribution, 226–8 ... Foundations, Evolution and Implementation Modeling and Valuation of Energy Structures Analytics, Econometrics, and Numerics Daniel Mahoney Director of Quantitative Analysis, Citigroup, USA ©... Library of Congress To Cathy, Maddie, and Jack Contents List of Figures List of Tables Preface Acknowledgments Synopsis of Selected Energy Markets and Structures 1.1 Challenges of modeling in energy. .. DERIVATIVES AND HYBRIDS Markets, Models and Methods Enrico Edoli, Stefano Fiorenzani and Tiziano Vargiolu OPTIMIZATION METHODS FOR GAS AND POWER MARKETS Theory and Cases Roland Lichters, Roland Stamm and